The real question is why does my cat not sleep when I do? At three in the morning, he leaps on me (in bed) and wants to play, go outside, annoy me and have a snack. His food dish is filled before I turn in, but apparently that’s not good enough.

Then in the AM, I am up and about, and he is curled into that C shape, apparently in a coma.

1) Like any small creature, they radiate more heat for their size than larger animals. That means that they have to either eat far more (as a percentage of their body mass) and/or reduce their activity level to conserve energy.
Many small animal spend practically every waking moment eating, but cats generally have other things to do; hunt, groom, play, secure their turf, etcetera. That means that they are opting for the second choice; conserving energy. That means more naps.

2) Cats are more nocturnal than humans. They do not have the same circadian rhythm that us humans do, so they are active at all hours, especially dusk and dawn. Unlike us humans, they are far from night-blind so time of day has less effect on their activity cycle than it does for most species.

@hiphiphopflipflapflop: Now that Milo has caught every available small rodent in the neighborhood (including inside my house) and the weather is really cold and raw, it is hard to amuse him.

He is happiest outdoors but loathes the snow and ice and now goes in and out in the same motion. I am finding it hard to play with him when I feel like it. He hates his toys except for the fishing rod with the string attached. I have no idea what he does while I am sleeping. There is a lot of staring at nothing during the day, however.

What @jerv said. Cats sleep more than almost any other mammal, generally up to 16 hours a day. Cats are “crepuscular,” which means they are most active at dawn and dusk. They do their hunting at these times because their prey is most active then. During the heat of the day, the cat is asleep. Although they have good night vision, cats can’t see in complete darkness, so they tend to sleep then as well.

Many house cats do sleep at night and have more waking hours during the daytime when their humans are up and about. However, you can easily convince a cat to stay up at night and sleep during the day if 1) you leave it home alone all day, and/or 2) when it does get you up in the middle of the night you reward it by feeding it, petting or paying attention to it.

The staring at nothing that cats do so much of is actually the cat concentrating on tiny sounds that we can’t hear while conserving energy for the time when a prey item comes out of hiding.

As someone else said, actually cats are magnificent mathematicians. When a cat is “staring at the wall” they are actually calculating the probability curve of every atom in the paint for fun and as the equivalent of mental weightlifting. (with apologies to Terry Pratchett)